Set cardinality, function onto, open unit square maps into real number set

Question

I have a question in my homework that I have trouble solving it. I'm not sure if I understand the question actually. I'll attach the question below and hope someone could give me any hints.

Consider the open interval $(0,1)$, and let $S$ be the set of points in the open unit square; that is, $S = \{ (x,y): 0 < x, y < 1 \}$. Use the fact that every real number has a decimal expansion to produce a $1-1$ function that maps $S$ into $(0,1)$. Discuss whether the formulated function is onto.

edited: there's a hint in the question "keep in mind that any terminating decimal expansion such as .235 represents the same real number as .2349999... ."

I think the "formulated function" is the $f: S \to (0,1)$. I don't know how to use the decimal expansion concept.

Thanks

Answer 1

Consider the two sequences of decimal expansions of $x$ and $y$. How do you combine them into one sequence injectively?

Answer 2

Do you know the extended problem of the Hilbert hotel? Suppose you have a hotel with infinitely many rooms numbered $R_1\cdots R_\omega$ which are all full. Suppose infinitely (but countably) many new guests arrive and you have to accommodate them all. The problem you have is very similar if you think of a point in $S=(a,b)$ as being the two sets of guests (the ones in the rooms already and the newly arrived ones) and the points in $(0,1)$ as each being a Hilbert hotel.You will ofcourse have to also deal with repeating decimals but you can manage that by just adding another set of guests to your hotel.

Answer 3

The idea is that if $\langle x,y\rangle\in S$, $x$ has the decimal expansion $0.d_1d_2d_3\ldots$, and $y$ has the decimal expansion $0.e_1e_2e_3\ldots$, we can mesh the expansions to send $\langle x,y\rangle$ to the real number $f(x,y)$ whose decimal expansion is $0.d_1e_1d_2e_2d_3e_3\ldots\;$. However, there are several problems that have to be dealt with.

First, as the hint points out, some real numbers have two decimal expansions: $0.5=0.4999\ldots$, for instance. If we want the function $f$ to be well-defined, we’ll have to specify which expansion to use for such numbers. For simplicity let’s agree to use the one that terminates, padded with zeroes, so that $\frac12$ will be represented by $0.5000\ldots$, not by $0.4999\ldots$. Thus, the point $\left\langle\frac13,\frac12\right\rangle$ is sent by $f$ to the real number whose decimal expansion is $0.35303030\ldots\;$.

You have to do two things. First, you have to prove that this map always produces a real number in $(0,1)$. Then you have to decide whether the function maps $S$ onto $(0,1)$ or only into $(0,1)$. In other words, is there some $x\in(0,1)$ that is not $f(x,y)$ for any $\langle x,y\rangle\in S$. HINT: Consider the number $\frac1{11}$.